stability of equilibria of nematic liquid crystalline...
TRANSCRIPT
Acta Mathematica Scientia 2011,31B(6):2289–2304
http://actams.wipm.ac.cn
STABILITY OF EQUILIBRIA OF NEMATIC
LIQUID CRYSTALLINE POLYMERS∗
Dedicated to Professor Peter D. Lax on the occasion of his 85th birthday
Hong Zhou
Department of Applied Mathematics
Naval Postgraduate School, Monterey, CA 93943, USA
E-mail: [email protected]
Hongyun Wang
Department of Applied Mathematics and Statistics
University of California, Santa Cruz, CA 95064, USA
E-mail: [email protected]
Abstract We provide an analytical study on the stability of equilibria of rigid rodlike
nematic liquid crystalline polymers (LCPs) governed by the Smoluchowski equation with
the Maier-Saupe intermolecular potential. We simplify the expression of the free energy
of an orientational distribution function of rodlike LCP molecules by properly selecting
a coordinate system and then investigate its stability with respect to perturbations of
orientational probability density. By computing the Hessian matrix explicitly, we are
able to prove the hysteresis phenomenon of nematic LCPs: when the normalized polymer
concentration b is below a critical value b∗ (6.7314863965), the only equilibrium state is
isotropic and it is stable; when b∗ < b < 15/2, two anisotropic (prolate) equilibrium states
occur together with a stable isotropic equilibrium state. Here the more aligned prolate
state is stable whereas the less aligned prolate state is unstable. When b > 15/2, there are
three equilibrium states: a stable prolate state, an unstable isotropic state and an unstable
oblate state.
Key words equilibria; stability; nematic liquid crystalline polymers; hysteresis phe-
nomenon
2000 MR Subject Classification 35Kxx; 70Kxx
1 Introduction
Since its discovery by Austrian botanical physiologist Friedrich Reinitzer in 1888, liquid
crystals have spurted intensive experimental, theoretical and numerical studies [1, 5, 6, 18, 31,
39]. One notable example is that in 1991 Pierre-Gilles de Gennes was awarded the Nobel Prize
∗Received September 30, 2011. This work is partially supported by the National Science Foundation and
by the Office of Naval Research.
2290 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B
in Physics “for discovering that methods developed for studying order phenomena in simple
systems can be generalized to more complex forms of matter, in particular to liquid crystals
and polymers”.
Liquid crystals refer to a state of matter that has properties between those of a solid crystal
and those of an isotropic liquid. For example, a liquid crystal may flow like an ordinary liquid,
but its molecules may be oriented in some directions. Examples of liquid crystals can be found
both in nature (e.g. solutions of soap and detergents) and in technological applications (e.g.
electronic displays). There are many different types of liquid crystal phases, among which one
of the most common phases is the nematic. In a nematic phase, the rod-shaped molecules have
no positional order but they display long-range directional order with their long axes roughly
aligned parallel to a common axis (which is called a “director”). Most nematics are uniaxial,
meaning that they have one axis that is longer and preferred while the other two axes are
equivalent. However, some nematics are biaxial which implies that the molecules also orient
along a secondary axis in addition to orienting their long axis. Nematics can be easily aligned
by applying an external magnetic or electric field. The optical properties of aligned nematics
make them extremely useful in liquid crystal displays.
Liquid crystallinity in polymers may occur either by heating a polymer above its melting
point (thermotropic LCPs) or by dissolving a polymer in a solvent (lyotropic LCPs). Liquid
crystalline polymers combine the properties of liquid crystals with the properites of polymers.
As a result, it is easy to guide LCPs by applying small external forces such as flows and it is
easy to control their chemical and material parameters. LCPs exhibit certain anisotropy of the
electrical, mechanical and magnetic properties with high flexibility or elasticity. They are also
exceptionally unreactive and inert, and highly resistant to fire. Due to their various properties,
LCPs have found more and more applications, including electrical and mechanical parts, food
containers, and any other applications requiring chemical inertness and super strength. A main
example of LCPs in solid form is high strength fibers which have been made into bullet-proof
vests and airbags.
The theoretical studies of liquid crystals traced back more than 60 years ago. In 1949
Onsager [29] developed a statistical theory showing that simple exclude-volume repulsions be-
tween long rods are sufficient to create a liquid crystal. Onsager’s theory is valid for dilute
liquid crystals. Flory [9, 10] presented a lattice model and his theory yields good approxima-
tion for dense and highly ordered liquid crystals. Both the Onsager and Flory theories predict
an isotropic-nematic phase transition when the concentration is high enough. In three pub-
lications Maier and Saupe [26–28] proposed a mean-field theory to explain the occurrence of
a liquid crystal phase in low molecular weight materials based on the assumption that only
induced dipolar forces are relevant for orientational ordering. An impressive signature for the
popularity of the Maier-Saupe theory is its high number of citations (almost 4000 up to now)
over the years [30]. In 1965 Landau and Ginzburg [22] developed a phenomenological theory for
liquid crystals. Even though it has success in most cases, Landau-Ginzburg theory is not exact
and it suffers failure in some cases [18]. One of the earliest constitutive theories of liquid crystals
is the Leslie-Ericksen vector theory [7, 23] in 1970s. The Leslie-Ericksen theory is attractive for
studying low molar-mass liquid crystals.
After Maier and Saupe and Flory’s work, many new theories have been developed for liquid
No.6 H. Zhou & H.Y. Wang: STABILITY OF EQUILIBRIA OF NEMATIC LIQUID 2291
crystalline polymers. One of the most famous kinetic models for polymer liquid crystals was
developed by Doi and Edwards [6] in 1980s and Hess [19]. The crucial idea of the Doi theory is to
model the polymeric molecules as a suspension of rigid rodlike nematogens and then describe the
ensemble with an orientational probability distribution function. The orientational distribution
function is governed by a nonlinear Smoluchowski (or Fokker-Planck) equation which takes into
account the hydrodynamic, Brownian and intermolecular forces.
In recent years, the Doi theory has attracted enormous mathematical attentions (for ex-
ample, [2–4, 8, 11–17, 21, 24, 25, 32–38]). One mathematical advance in the understanding
of the equilibrium states of the Doi model is the rigorous proof on that the equilibria of the
Smoluchowski equation with the Maier-Saupe potential are uniaxial [8, 24, 35]. However, it
seems more challenging to obtain a solid proof on the stability of the equilibria. The purpose
of this paper is to circumvent this difficulty and seek a mathematical proof on the stability of
the equilibria. The main result proven here is: all stable equilibria are either isotropic or highly
aligned prolate uniaxial.
2 Background and Formulation
For reader’s convenience, we start with the briefest glimpse of the Doi kinetic theory.
Let us denote the orientational direction of each rodlike molecule by a unit vector m,
the probability density function by ρ(m). The evolution of the probability density function is
described by the Smoluchowski equation [6]:
∂ρ
∂t= D
∂
∂m·
(1
kBT
∂U
∂mρ +
∂ρ
∂m
), (1)
where ∂/∂m is the orientational gradient operator [1], D the rotational diffusivity, kB the
Boltzmann constant, and T the absolute temperature. For simplicity, we set kBT = 1 which is
equivalent to normalizing U by kBT .
For nematic polymers, the total potential consists of only the Maier-Saupe interaction
potential
U(m) = −b〈mm〉 : mm, (2)
where the tensor product mm and tensor double contraction A : B are defined as
mm ≡
⎡⎢⎢⎣
m1m1 m1m2 m1m3
m2m1 m2m2 m2m3
m3m1 m3m2 m3m3
⎤⎥⎥⎦ ,
A : B ≡ trace(AB).
In (2) the constant b is proportional to the normalized polymer concentration and it describes
the strength of inter-molecular interactions, and 〈mm〉 denotes the second moment of the
orientation distribution:
〈mm〉 ≡
∫‖m‖=1
mm ρ(m) dm,
where ρ(m) is the orientational probability density function of the ensemble of polymer rods,
i.e., the probability density that a polymer rod has direction m.
2292 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B
The Maier-Saupe potential in a properly selected coordinate system:
Let us select a coordinate system such that the second moment 〈mm〉 is diagonal:
〈mm〉 =
⎡⎢⎢⎣〈m2
1〉 0 0
0 〈m22〉 0
0 0 〈m23〉
⎤⎥⎥⎦ =
⎡⎢⎢⎢⎢⎢⎢⎣
s1 +1
30 0
0 s2 +1
30
0 0 s3 +1
3
⎤⎥⎥⎥⎥⎥⎥⎦
,
where sj ≡ 〈m2j〉 −
13 satisfies
s1 + s2 + s3 = 〈m21 + m2
2 + m23〉 − 1 = 0.
In this coordinate system, the Maier-Saupe potential takes the following expression:
U(m) = −b(〈m21〉m
21 + 〈m2
2〉m22 + 〈m2
3〉m23)
= −b
3− b(s1m
21 + s2m
22 + s3m
23). (3)
The constant (−b/3) does not affect the dynamics or the stability of the nematic polymer
ensemble. For simplicity, we drop the constant (−b/3). From s1+s2+s3 = 0 and m21+m2
2+m23 =
1, we have
s1 =−s3
2−
s2 − s1
2,
s2 =−s3
2+
s2 − s1
2,
m21 + m2
2 = 1−m23.
Using these equations, we can rewrite the Maier-Saupe potential (3) as
U(m) = −b
((−s3
2−
s2 − s1
2
)m2
1 +
(−s3
2+
s2 − s1
2
)m2
2 + s3m23
)
= −b3 s3
2
(m2
3 −1
3
)− b
(s2 − s1)
2(m2
2 −m21)
= −η1
(m2
3 −1
3
)− η2(m
22 −m2
1), (4)
where
η1 ≡ b3 s3
2= b
3
2
(〈m2
3〉 −1
3
), (5)
η2 ≡ b(s2 − s1)
2= b
1
2〈m2
2 −m21〉. (6)
Note that for any orientational distribution ρ(m), by selecting a proper coordinate system, we
can always express the Maier-Saupe potential U(m) in the form of (4) with η1 and η2 given by
(5) and (6). In particular, ρ(m) does not have to be an equilibrium distribution.
The Boltzmann distribution:
No.6 H. Zhou & H.Y. Wang: STABILITY OF EQUILIBRIA OF NEMATIC LIQUID 2293
In the absence of external field, the equilibrium distributions of a nematic polymer ensemble
are described by the Boltzmann distribution [6]:
ρ(BZ)(m; η1, η2) =exp[−U(m)]∫
Sexp[−U(m)] dm
=exp[η1m
23 + η2(m
22 −m2
1)]∫
Sexp [η1m2
3 + η2(m22 −m2
1)] dm, (7)
where S represents the unit sphere. It is clear that an equilibrium distribution is completely
specified by (η1, η2). To find an equilibrium distribution, we need to determine (η1, η2). The
governing equation for (η1, η2) is obtained by combining the definition of (η1, η2) and the Boltz-
mann distribution (7):
η1 = b3
2
∫S
(m2
3 −1
3
)ρ(BZ)(m; η1, η2) dm,
η2 = b1
2
∫S
(m2
2 −m21
)ρ(BZ)(m; η1, η2) dm. (8)
Note that while the Maier-Saupe potential (4) is always true, the Boltzmann distribution (7) is
valid only for equilibrium distributions. It is very important to notice this difference between
(4) and (7) especially when we consider the free energy of orientational distributions in the
stability discussion.
Equilibrium states of a nematic polymer ensemble:
It has been shown that in the absence of external field, all equilibrium states of nematic
polymers are axisymmetric [8, 24, 35]. Without loss of generality, we assume the m3-axis is the
axis of symmetry. The axisymmetry implies s1 = s2 and consequently we have η2 = 0. Thus,
an equilibrium state of nematic polymers is completely specified by η1.
To facilitate the discussion, we use r(b) to denote the value of η1 in the equilibrium state(s)
corresponding to parameter b. More specifically, we adopt the convention that η, η1 and η2
denote general variables while r(b) denotes specific value of η1 in the equilibrium state(s).
By definition r(b) is the solution of equation (8). With η2 = 0 and η1 simply denoted by
η, equation (8) becomes
η = b3
2
∫S
(m2
3 −13
)exp(η m2
3
)dm∫
Sexp (η m2
3) dm. (9)
It is now convenient to introduce the function f(η) defined by
f(η) ≡3
2η
∫S
(m2
3 −13
)exp(η m2
3
)dm∫
Sexp (η m2
3) dm. (10)
Then it follows at once that equation (9) can be written as [35]
η
(1
b− f(η)
)= 0. (11)
First, observe that η = r0(b) = 0 is always a solution of equation (11), which corresponds to
U(m) = constant (independent of m), the isotropic equilibrium. Next, anisotropic equilibrium
states are the solutions of the equation
f(η) =1
b. (12)
2294 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B
Using the spherical coordinates,
m1 = sin φ cos θ,
m2 = sin φ sin θ,
m3 = cosφ, (13)
followed by the substitution u = cosφ and then applying integration by parts, we write function
f(η) as
f(η) =3
2η
∫ π
0
(cos2 φ− 1
3
)exp(η cos2 φ
)sin φdφ∫ π
0exp (η cos2 φ) sin φdφ
=3
2η
∫ 1
−1
(u2 − 1
3
)exp(η u2)du∫ 1
−1exp (η u2) du
=
∫ 1
−1
(u2 − u4
)exp(η u2)du∫ 1
−1exp (η u2) du
=
∫S
(m2
3 −m43
)exp(η m2
3
)dm∫
Sexp (η m2
3) dm. (14)
In [35], we have shown that the function f(η) expressed in (14) satisfies the following properties:
1. f(0) = 215 ;
2. limη→+∞
f(η) = 0 and limη→−∞
f(η) = 0; in other words, f(η) tends to zero as |η| goes to
infinity.
3. f(η) attains its maximum at η∗ = 2.1782879748 > 0 where the maximum is f(η∗) =
0.14855559992254 > 0;
4. f ′(η) > 0 for η < η∗ and f ′(η) < 0 for η > η∗.
The graph of the function f(η) is depicted in Figure 1.
Fig. 1 Graph of the function f(η). A key feature of the function f(η) is that f(η) is strictly
increasing for η < η∗ (i.e. f ′(η) > 0) and strictly decreasing for η > η∗ (i.e., f ′(η) < 0)
From these properties of the function f(η), we draw conclusions listed below related to
equation (12):
• b∗ = 1f(η∗) = 6.7314863965 is a critical value for parameter b.
No.6 H. Zhou & H.Y. Wang: STABILITY OF EQUILIBRIA OF NEMATIC LIQUID 2295
• For b < b∗, equation (12) has no solution; that is, there is no anisotropic state for b < b∗.
• At b = b∗, equation (12) has one solution r(b∗) = η∗ > 0.
• For b∗ < b < 152 , equation (12) attains two solutions: rU (b) > η∗ > 0 and 0 < rM (b) <
η∗. The equilibrium states corresponding to positive order parameters rU (b) and rM (b) are
called prolate states.
• At b = 152 , equation (12) possesses two solutions: rU ( 2
15 ) > η∗ > 0 and r( 215 ) = 0.
• For b > 152 , equation (12) has two solutions: rU (b) > η∗ > 0 and rL(b) < 0. The
equilibrium state with negative order parameter rL(b) is called oblate state.
Here we use the subscript “U” to refer to the “Upper” part of the phase diagram where r > η∗,
the subscript “M” to represent the “Middle” part of the phase diagram where 0 < r < η∗, and
the subscript “L” to denote the “Lower” part of the phase diagram where r < 0. The phase
diagram for nematic polymers is shown in Figure 2.
Fig. 2 Phase diagram of nematic polymers
In the above, we have used b as the independent variable and treated r as a function of
b. However, r(b) is a multi-valued function of b and for b < b∗ the function r(b) is not even
defined. When we study the relation between b and r, it is mathematically more convenient
if we use r as the independent variable and treat b as a function of r instead. Then, b(r) is a
single-valued function of r and is defined for all values of r in (−∞, +∞). The function b(r)
can be easily determined from equation (12) as
b(r) =1
f(r). (15)
Once we know the function b(r), the branch rU (b) is simply the inverse function of b(r) for
r > η∗; the branch rM (b) is the inverse function of b(r) for 0 < r < η∗; and the branch rL(b) is
the inverse function of b(r) for r < 0.
Free energy of an orientational distribution:
The free energy of the orientational distribution ρ(m) is
G([ρ]) =
∫S
[log ρ(m) +
1
2U(m, [ρ])
]ρ(m)dm. (16)
Recall that for any ρ(m), by selecting a proper coordinate system, we can always write the
2296 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B
Maier-Saupe potential U(m, [ρ]) as
U(m, [ρ]) = −η1
(m2
3 −1
3
)− η2(m
22 −m2
1),
where η1 ≡ b 32
(〈m2
3〉 −13
)and η2 ≡ b 1
2 〈m22 −m2
1〉. But in general, ρ(m) is not the same as the
Boltzmann distribution
ρ(m) �= ρ(BZ)(m; η1, η2) ≡exp[η1m
23 + η2(m
22 −m2
1)]∫
Sexp [η1m2
3 + η2(m22 −m2
1)] dm.
It is important to point out that even if we restrict our consideration to probability density
ρ(m) that has the form of
ρ(m) =exp[q1m
23 + q2(m
22 −m2
1)]∫
Sexp [q1m2
3 + q2(m22 −m2
1)] dm,
we still have (q1, q2) �= (η1, η2) unless ρ(m) is an equilibrium distribution, which means q2 = 0
and q1 = r(b). It is tempting to replace ρ(m) by ρ(BZ)(m; η1, η2) in (16) and formally define a
function of (η1, η2)
F (η1, η2) ≡
∫S
[log ρ(BZ)(m; η1, η2) +
1
2
(−η1
(m2
3 −1
3
)− η2(m
22 −m2
1)
)]×ρ(BZ)(m; η1, η2)dm.
Unfortunately, the assertion F (η1, η2) = G([ρ]) is valid only when ρ(m) is an equilibrium
distribution. In other words, at a fixed value of b, the assertion F (η1, η2) = G([ρ]) is true
only at a few (at most, three) equilibrium states that are well separated from each other.
Therefore, while we can try to analyze the stability of F (η1, η2) with respect to perturbations
in (η1, η2), the stability of F (η1, η2) does not inform us about the stability of G([ρ]) with respect
to perturbations in ρ(m).
3 Stability Analysis
In this section we present a detailed stability analysis on the equilibria.
To make our presentation easy to follow, we divide our approach into five steps. In the first
step, we minimize the free energy functional G([ρ]) over a set B of functions with the second
moments fixed, specified by two parameters (η1, η2). The resulting free energy is a function of
two variables G(η1, η2). The stability of G([ρ]) is equivalent to the stability of G(η1, η2). In the
second step, we show that the minimum of G([ρ]) is attained at a probability density ρ∗ of the
Boltzmann form. The coefficients (q1, q2) in the exponent of ρ∗ define a one-to-one mapping
between (η1, η2) and (q1, q2). Thus, we can rewrite G(η1, η2) as H(q1, q2). The important point
to note here is that (q1, q2) is in general different from (η1, η2) unless ρ∗ is an equilibrium
state, which is an improper assumption in stability analysis. In the third step, we compute the
Hessian matrix of H(q1, q2) with respect to (q1, q2), which is fortunately a diagonal matrix. In
the fourth step, we consider the stability of the isotropic equilibrium state. At the last step we
focus our attention on the stability of the anisotropic equilibrium states.
No.6 H. Zhou & H.Y. Wang: STABILITY OF EQUILIBRIA OF NEMATIC LIQUID 2297
Step 1 We consider the constrained minimum over the set
B(η1, η2) =
{ρ(m)
∣∣∣∣(〈m2
3〉[ρ] −1
3
)=
2
3bη1,⟨m2
2 −m21
⟩[ρ]
=2
bη2
},
where the probability density ρ(m) is involved in the average in the following way:
〈f(m)〉[ρ] ≡
∫S
f(m)ρ(m)dm.
The minimum of G([ρ]) over the set B(η1, η2) is a function of (η1, η2):
G(η1, η2) ≡ minρ(m)∈B(η1,η2)
G([ρ]). (17)
From the definition of G(η1, η2), we see that the stability of G([ρ]) with respect to ρ is the same
as the stability of G(η1, η2) with respect to (η1, η2).
Step 2 We study where G([ρ]) attains the minimum over the set B(η1, η2) and use the
result to calculate G(η1, η2). For ρ(m) ∈ B(η1, η2), we have
U(m, [ρ]) = −η1
(m2
3 −1
3
)− η2(m
22 −m2
1),
1
2
∫S
U(m, [ρ])ρ(m)dm =1
2
(−η1
⟨m2
3 −1
3
⟩[ρ]
− η2〈m22 −m2
1〉[ρ]
)
= −
(1
3bη21 +
1
bη22
).
That is, the second term in the free energy G([ρ]) given in (16) is actually a constant over the
set B(η1, η2). It follows that we only need to look at the first term of G([ρ]) in the constrained
minimization problem. Let us denote the first term by
G1([ρ]) ≡
∫S
log ρ(m)ρ(m)dm. (18)
The constrained minimization problem then becomes
argminρ(m)∈B(η1,η2)
G1([ρ]).
Suppose the constrained minimum is attained at ρ∗(m). We consider the perturbed probability
density ρ(m) = ρ∗(m) + s Δρ(m) where Δρ(m) satisfies the following properties:∫S
Δρ(m)dm = 0,∫S
(m2
3 −1
3
)Δρ(m)dm = 0,∫
S
(m22 −m2
1)Δρ(m)dm = 0. (19)
G1([ρ∗ + s Δρ]) attaining the minimum at s = 0 implies dG1([ρ
∗+s Δρ])ds
∣∣∣s=0
= 0, which gives us
∫S
log ρ∗(m)Δρ(m)dm = 0. (20)
2298 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B
Since equation (20) is true for all Δρ(m) satisfying (19), it implies that
log(
argminρ(m)∈B(η1,η2)
G1([ρ]))
= log ρ∗(m) = q0 + q1
(m2
3 −1
3
)+ q2(m
22 −m2
1). (21)
The result of the constrained minimization given in (21) defines a mapping from (η1, η2) to
(q1, q2). The mapping from (η1, η2) to (q1, q2) is not expressed in an explicit form and it is
not immediately obvious if the mapping is single-valued. However, the inverse mapping from
(q1, q2) back to (η1, η2) is single-valued and has a simple explicit expression:
η1 = b3
2
⟨m2
3 −1
3
⟩,
η2 = b1
2〈m2
2 −m21〉, (22)
where the average is with respect to the probability density ρ∗(m, q1, q2) given by
ρ∗(m, q1, q2) ≡1
Z(q1, q2)exp
[q1
(m2
3 −1
3
)+ q2(m
22 −m2
1)
], (23)
Z(q1, q2) ≡
∫S
exp
[q1
(m2
3 −1
3
)+ q2(m
22 −m2
1)
]dm. (24)
To prepare for the computation of the Hessian matrix, we prove the following lemma,
which asserts that the mapping from (η1, η2) to (q1, q2) is also single-valued.
Lemma 1 For any fixed (η1, η2), there is only one set of (q1, q2) satisfying (22).
Proof We prove by contradiction. To accomplish this, suppose (q(1)1 , q
(1)2 ) and (q
(2)1 , q
(2)2 )
are different from each other, and both satisfy (22).
Let ρ∗(m) ≡ ρ∗(m, q(1)1 , q
(1)2 ) and Δρ(m) ≡ ρ∗(m, q
(2)1 , q
(2)2 )− ρ∗(m, q
(1)1 , q
(1)2 ).
We calculate the first and second derivatives of the function G1([ρ∗ + s Δρ]) with respect to s
as follows:
dG1([ρ∗ + s Δρ])
ds=
∫S
log(ρ∗ + s Δρ)Δρ dm,
d2G1([ρ∗ + s Δρ])
ds2=
∫S
1
(ρ∗ + s Δρ)(Δρ)2dm > 0.
Since both (q(1)1 , q
(1)2 ) and (q
(2)1 , q
(2)2 ) satisfy (22), Δρ(m) must satisfy (19), which means
dG1([ρ∗ + s Δρ])
ds
∣∣∣∣s=0
=
∫S
log(ρ∗)Δρdm = 0.
Taylor’s theorem tells us
G1([ρ∗ + Δρ]) = G1([ρ
∗]) +1
2
d2G1([ρ∗ + sΔρ])
ds2
∣∣∣∣s=ξ
> G1([ρ∗]),
which is
G1([ρ∗(m, q
(2)1 , q
(2)2 )]) > G1([ρ
∗(m, q(1)1 , q
(1)2 )]). (25)
Repeating the argument with the roles of (q(1)1 , q
(1)2 ) and (q
(2)1 , q
(2)2 ) swapped, we get
G1([ρ∗(m, q
(1)1 , q
(1)2 )]) > G1([ρ
∗(m, q(2)1 , q
(2)2 )]), (26)
No.6 H. Zhou & H.Y. Wang: STABILITY OF EQUILIBRIA OF NEMATIC LIQUID 2299
which contradicts with (25). Thus the claim is established: (q(1)1 , q
(1)2 ) and (q
(2)1 , q
(2)2 ) must be
the same. �
We conclude that the mapping between (η1, η2) and (q1, q2) is one-to-one. As a result, we
can express the minimum free energy over the set B(η1, η2) as a function of (q1, q2):
H(q1, q2) ≡ minρ(m)∈B(η1,η2)
G([ρ])
= q1
⟨m2
3 −1
3
⟩+ q2
⟨m2
2 −m21
⟩− log(Z(q1, q2))−
1
b
(1
3η21 + η2
2
)
=2
b
(1
3q1η1 + q2η2
)− log(Z(q1, q2))−
1
b
(1
3η21 + η2
2
), (27)
where the mapping from (q1, q2) to (η1, η2) is given in (22) and equations (16), (23) and (24)
have been invoked.
The stability of G([ρ]) with respect to ρ(m) is the same as the stability of H(q1, q2) with
respect to (q1, q2), which is determined by its Hessian matrix. The Hessian matrix (or simply
the Hessian) is just the square matrix of second-order partial derivatives of a function. It can
be used to determine the stability or instability of a system.
Step 3 We now calculate the Hessian matrix of H(q1, q2). To do so, we begin by
differentiating η1, η2 and log(Z(q1, q2)) with respect to q1 and q2. Referring to (22), (23) and
(24) and by direct computation, we obtain the following partial derivatives:
∂ρ∗(m, q1, q2)
∂q1=
(m2
3 −1
3
)ρ∗(m, q1, q2)−
⟨m2
3 −1
3
⟩ρ∗(m, q1, q2)
= (m23 − 〈m
23〉)ρ
∗(m, q1, q2),
∂ρ∗(m, q1, q2)
∂q2=((m2
2 −m21)− 〈m
22 −m2
1〉)ρ∗(m, q1, q2),
∂η1
∂q1= b
3
2
(〈m4
3〉 − 〈m23〉
2)
> 0,
∂η1
∂q2= b
3
2
(〈m2
3(m22 −m2
1)〉 − 〈m23〉〈m
22 −m2
1〉),
∂η2
∂q1= b
1
2
(〈m2
3(m22 −m2
1)〉 − 〈m23〉〈m
22 −m2
1〉),
∂η2
∂q2= b
1
2
(〈(m2
2 −m21)
2〉 − 〈m22 −m2
1〉2)
> 0,
∂ log(Z(q1, q2))
∂q1=
⟨m2
3 −1
3
⟩=
2
3bη1,
∂ log(Z(q1, q2))
∂q2=⟨m2
2 −m21
⟩=
2
bη2.
Applying these results to differentiate H(q1, q2) given in (27), we get
∂H(q1, q2)
∂q1=
2
b
(1
3(q1 − η1)
∂η1
∂q1+ (q2 − η2)
∂η2
∂q1
),
∂H(q1, q2)
∂q2=
2
b
(1
3(q1 − η1)
∂η1
∂q2+ (q2 − η2)
∂η2
∂q2
).
2300 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B
To determine the stability of an equilibrium state, we only need to calculate the Hessian matrix
of H(q1, q2) evaluated at the equilibrium state.
At an equilibrium state, (η1, η2)|eq satisfies equation (8). The mapping between (q1, q2)
and (η1, η2) is described in (22). Comparing (8) and (22), we conclude
(q1, q2)|eq = (η1, η2)|eq .
In addition, all equilibrium states are axisymmetric, which amounts to
q2|eq = 0.
Thus, at an equilibrium state, we have
〈m22 −m2
1〉∣∣eq
= 0,
〈m23(m
22 −m2
1)〉∣∣eq
= 0,
which leads to
∂η1
∂q2
∣∣∣∣eq
= 0,
∂η2
∂q1
∣∣∣∣eq
= 0.
Calculating the Hessian matrix of H(q1, q2) and evaluating it at the equilibrium state, we obtain
∂2H(q1, q2)
∂q21
∣∣∣∣eq
=2
3b
(1−
∂η1
∂q1
)∂η1
∂q1
∣∣∣∣eq
,
∂2H(q1, q2)
∂q1∂q2
∣∣∣∣eq
= 0,
∂2H(q1, q2)
∂q22
∣∣∣∣eq
=2
b
(1−
∂η2
∂q2
)∂η2
∂q2
∣∣∣∣eq
.
Note that both ∂η1
∂q1
and ∂η2
∂q2
are positive. Consequently, the stability of H(q1, q2) is determined
by the two quantities below:
H11 ≡ 1−∂η1
∂q1
∣∣∣∣eq
= 1− b3
2
(〈m4
3〉 − 〈m23〉
2)∣∣∣∣
eq
,
H22 ≡ 1−∂η2
∂q2
∣∣∣∣eq
= 1− b1
2〈(m2
2 −m21)
2〉
∣∣∣∣eq
.
Since q2|eq = 0, we can write 〈(m22 −m2
1)2〉|eq as
〈(m22 −m2
1)2〉|eq =
∫S(m2
2 −m21)
2 exp(q1m
23
)dm∫
Sexp (q1m2
3) dm
=
∫ π
0 (sin2 φ)2(
12π
∫ 2π
0 (cos2 θ − sin2 θ)2dθ)
exp(q1 cos2 φ) sin φdφ∫ π
0exp(q1 cos2 φ) sin φdφ
=1
2〈(1−m2
3)2〉
∣∣∣∣eq
.
No.6 H. Zhou & H.Y. Wang: STABILITY OF EQUILIBRIA OF NEMATIC LIQUID 2301
Thus, we write H22 as
H22 = 1− b1
4〈(1−m2
3)2〉
∣∣∣∣eq
.
Step 4 Now we turn our attention to the stability of the isotropic state, which is
described by (q1, q2)|eq = (0, 0). With the help of the spherical coordinates (13), it is straight-
forward to derive
〈m23〉|eq =
∫ π
0cos2 φ sin φdφ∫ π
0 sin φdφ=
∫ 1
0
u2du =1
3,
〈m43〉|eq =
∫ 1
0
u4du =1
5,
〈(1−m23)
2〉|eq =
∫ 1
0
(1− u2)2du =8
15,
H11 = 1− b2
15,
H22 = 1− b2
15.
It follows readily that the Hessian matrix evaluated at the isotropic equilibrium state is positive
definite if b < 152 and negative definite if b > 15
2 . Therefore, the isotropic equilibrium state is
stable for b < 152 whereas it is unstable for b > 15
2 .
Step 5 Finally, we close our analysis by addressing the stability of anisotropic equilib-
rium states. An anisotropic equilibrium state is described by q1 = r(b) and q2 = 0. First we
look at H11. We use the definition of function f(η) given in (10) to write
qf(q) =3
2
∫S
(m2
3 −13
)exp(q m2
3
)dm∫
Sexp (q m2
3) dm=
3
2
⟨m2
3 −1
3
⟩∣∣∣∣(q1,q2)=(q,0)
. (28)
Note that (28) is true for all values of q, not just at the equilibrium state q = r(b). Differentiating
(28) with respect to q and then evaluating it at the equilibrium state, we have
f(r(b)) + r(b)f ′(r(b)) =3
2
(〈m4
3〉 − 〈m23〉
2)∣∣∣∣
eq
. (29)
Since r(b) is the solution of equation (12), it satisfies f(r(b)) = 1/b. Substituting this into (29)
yields
H11 = 1− b3
2
(〈m4
3〉 − 〈m23〉
2)∣∣∣∣
eq
= −b r(b)f ′(r(b)).
Recall that in [35] we have derived that
f ′(η) < 0 for η > η∗ and f ′(η) > 0 for η < η∗
and recall the classification of three anisotropic branches
rU (b) > η∗, 0 < rM (b) < η∗, and rL(b) < 0,
we see that
H11 =
⎧⎪⎪⎨⎪⎪⎩
> 0 for rU (b),
< 0 for rM (b),
> 0 for rL(b).
(30)
2302 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B
Next we examine H22. Multiplying H22 by 4/b, using 1/b = f(r(b)) and the expression of f(η)
given in (14), we have
4
bH22 = 4f(r(b))− 〈(1−m2
3)2〉∣∣(q1,q2)=(r(b),0)
= 4⟨m2
3 −m43
⟩∣∣(q1,q2)=(r(b),0)
− 〈(1−m23)
2〉∣∣(q1,q2)=(r(b),0)
=⟨(1−m2
3)(5m23 − 1)
⟩∣∣(q1,q2)=(r(b),0)
≡ h(r(b)),
where the function h(q) is defined as
h(q) ≡⟨(1−m2
3)(5m23 − 1)
⟩∣∣(q1,q2)=(q,0)
. (31)
To determine the sign of h(r(b)), we study the behavior of function h(q). We start by verifying
h(0) =
∫ 1
0
(1 − u2)(5u2 − 1)du = 0.
Next we differentiate h(q):
h′(q) =⟨(1−m2
3)(5m23 − 1)(m2
3 − 〈m23〉)⟩∣∣
(q1,q2)=(q,0)
=1
5
⟨(1 −m2
3)(5m23 − 1)(5m2
3 − 1 + 1− 5〈m23〉)⟩∣∣
(q1,q2)=(q,0)
=1
5
⟨(1 −m2
3)(5m23 − 1)2
⟩∣∣(q1,q2)=(q,0)
+1
5(1− 5〈m2
3〉)h(q). (32)
Note that the first term in h′(q) is always positive for all values of q:⟨(1−m2
3)(5m23 − 1)2
⟩∣∣(q1,q2)=(q,0)
> 0.
As a result, the function h(q) has the property that h′(q) > 0 wherever h(q) = 0. This property
along with h(0) = 0 implies
h(q) > 0 for q > 0 and h(q) < 0 for q < 0.
Thus, for H22, we have
H22 =b
4h(r(b)) =
⎧⎪⎪⎨⎪⎪⎩
> 0 for rU (b),
> 0 for rM (b),
< 0 for rL(b).
(33)
Putting the results from (30) and (33) together, we arrive at
(H11, H22) =
⎧⎪⎪⎨⎪⎪⎩
(+, +) for rU (b),
(−, +) for rM (b),
(+,−) for rL(b).
Therefore, only the highly aligned prolate branch rU (b) is stable while the weakly aligned prolate
branch rM (b) and the oblate branch rL(b) are unstable. More specifically, the branch rU (b) is
stable to all perturbations; the branch rM (b) is unstable with respect to perturbation in q1,
i.e., axisymmetric perturbation along the axis of symmetry. Such perturbation will drive the
system either to highly aligned prolate branch rU (b) or the isotropic branch. In contrast, the
oblate branch rL(b) is unstable with respect to perturbation in q2, i.e., perturbation away from
axisymmetry. This kind of perturbation will carry the system to the prolate state.
No.6 H. Zhou & H.Y. Wang: STABILITY OF EQUILIBRIA OF NEMATIC LIQUID 2303
4 Conclusions and Perspective Views
The stability analysis performed in this study confirms the well-known hysteresis phe-
nomenon of the equilibria of the nematic liquid crystalline polymers governed by the Smolu-
chowski equation with the Maier-Saupe intermolecular potential. In particular, the isotropic
equilibrium state is stable for b < 15/2 and it becomes unstable for b > 15/2 where b is
proportional to the normalized polymer concentration and it describes the strength of the in-
termolecular interactions. As for the anisotropic equilibrium states, only the highly aligned
prolate state is stable while the less aligned prolate state (b∗ < b < 15/2) and the oblate state
(b > 15/2) are unstable. The anisotropic states occur when b > b∗ where the critical point
b∗ = 6.7314863965.
While the mathematical study on the Smoluchowski equation with the Maier-Saupe po-
tential is getting well-developed, rigorous proofs on the similiar results for the Smoluchowski
equation with the Onsager potential remain to be explored. We end this paper by adopting a
perspective view provided by Donald, Windle and Hanna [5]: “It is clear that ‘self-assembly’,
where molecules are designed so that they organise themselves into larger scale structures in
order to achieve special properties, will be a significant objective in materials science in the
twenty-first century. The demands of nanotechnology require an ever finer control over the
molecular arrangements within new materials. The fact that liquid crystallinity itself is a form
of orientational self-assembly, coupled with the fact that the molecules in a mesophase can be
steered by external fields, means that the principles undering the science of LCPs can only grow
in significance in the years ahead.”
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